Integrals related to stationary processes and cylindrical martingales

A NNALES DE L ’I. H. P., SECTION B

J. P ELLAUMAIL

A. W ^ERON

Integrals related to stationary processes and cylindrical martingales

Annales de l’I. H. P., section B, tome 15, n

^o

2 (1979), p. 127-146

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Integrals ^{related to} stationary processes and cylindrical martingales

J. PELLAUMAIL and A. WERON Ann. Inst. Henri Poincaré,

Vol. XV, ^n° 2, 1979, p. 127 -146.

Section B : Calcul des Probabilités et Statistique.

SUMMARY . The aim of this _paper is to

analyse

^a

relationship

^between

integrals

^{related to}

stationay

processes and _square

integrable martingales.

We consider the case when processes take their values in Banach _spaces.

RESUME.

2014 L’objet

^{de ce travail est de mettre en} evidence Fctroitc cor-

relation existant entre diverses

integrales qui

interviennent dans

1’etude,

d’une

part,

^des processus

stationnaires,

^d’autre

part,

^des

martingales cylindriques.

TABLE OF CONTENTS

0. Introduction ... 128

1. Quadratic integral ... 128

2. Non-negative ^measures ... 130

3. Operator valued functions and quadratic integral ^{... 134}

4. Hilbert space ^case ... 136

5. Stochastic integral ... 138

6. Loynes spaces ... 140

7. Stationary processes ... 142

8. Square integrable cylindrical martingales ... 144

References ... 145

Annales de l’lnstitut Henri Poincare-Section B-Vol. XV, 0020-2347/1979/127i $ 4,00 0

127

INTRODUCTION

The aim of this _paper is to

analyse

^a

relationship

^between

integrals

related to

stationary

processes and _square

integrable martingales.

^We

consider the case when processes take their values in Banach spaces.

We establish here that

integrals (stochastic

^and

deterministic)

^{used for}

the above two classes of stochastic _processes are

closely

^{related. First}

we discuss such called

quadratic integral

^{for vector} valued functions and for their tensor

products.

^{After we direct our attension at}

non-negative

measures with values in the space

(B’ @ 1 B’)’,

^{where B is a} Banach space.

In sections 3. 5 two

types

^of

integrals

^for

operator

^{valued functions are} described.

Finally,

in the last sections 7 and 8 ^we show how the above

general approach

^{is used for}

stationary

processes and for _square

integrable cylin-

drical

martingales.

In the paper ^{we use} the

langage

^{of tensor}

products

^{and the dilation}

theory,

which has been

recently developed

^for

non-negative

^{B-to-B’ valued} ope- rators. We

hope

^{that this}

gives

^{a new}

light

^for

problems

^{discussed here.}

The authors would like to express their

gratitude

^{to Professor M. Meti-}

vier for _many valuable

suggestions

^and conversations.

1

QUADRATIC

^INTEGRAL

In this first

section,

^{we recall some} definitions and results on

integration

with

respect

^{to vector measures and on tensor}

product:

^we

apply

^this

notions to the

study

^of

quadratic integrals.

1.1. BILINEAR INTEGRAL.

We consider

- a set S and a

a-algebra L

^{of subsets}

^{of S,}

- a

complex

Banach space ^V and its

topological

^dual

V’,

- an additive V’-valued function M defined on

E,

- a V-valued function

f

^{defined on S.}

The first

problem

^{is to} define the

integral ^fdM.

^{For an extensive}

study

^{on this}

problem,

^{we refer to}

[Bar ] .

^~

Annales de l’Institut Henri Poincaré - Section B

129 INTEGRALS RELATED TO STATIONARY PROCESSES

It is convenient for our purpose ^to consider the total variation which is defined as follows : if A

belongs

^{to L}

where this _supremum is taken over all the finite L-measurable

partitions

of A.

In

general situation,

this total variation is not finite

(see 2.7),

^{but in}

the

following

^we often will _suppose that this total variation _vM is finite

(or a-finite).

If the total variation _vM is finite and

6-additive,

^{M is}

strongly

^a-additive

dM and there exists a V’-valued weak

Radon-Nikodym

derivative M’ =

-d

of M with

respect

^{to this R. N.} derivative is V’-valued and for each element v of

V, ( M’, v ~

is the classical real R. N. derivative

of ( M, v ~

with

respect

^to

(briefly

^{w. r.}

t.)

^vM.

Conversely,

^{if this R. N.} derivative M’

exists w. r. t. a

positive

^measure ,u, then the total variation _vM is finite

(see [Pel]).

In the

following,

^{when we} say M is a measure, it means that M is a a-addi- tive measure.

For the convenience we note the

following elementary

^{lemma :}

1.2. LEMMA.

Let M be a function defined on an

algebra

^{d and with values in the dual V’}

of a Banach space V. We suppose that the total variation _’!M of M is finite and

that,

for each element v

of V,

^{the real}

function ~ v, M(.) )

is (7-additive.

Then the total variation _vM is a-additive.

The

proof

^{is left to the reader.}

1.3. TENSOR PRODUCT.

Let B and C be two

locally

^{convex vector} spaces :

sometimes,

^{we have}

to use a bilinear continuous

mapping g

^{defined on B x C :}

often,

^{it is}

convenient to consider this bilinear

mapping

^{as a linear}

mapping

^defined

on the « tensor

product »

^B

(x)

^C.

For definitions and

properties

^{of tensor}

product,

^{we refer to}

[Tre].

Let ^us recall

only

^some

properties

when B and C are Banach spaces. Let D be a Banach space, B

Q 1 C

^{is a Banach} space such

that,

^for each D-valued bilinear continuous

mapping g

^{defined on B x}

C,

^{there exists a D-valued}

J. PELLAUMAIL

linear

mapping f

^{defined on B}

@ 1 C

^{such that the}

following diagram

^is

commutative

where j

^{is a canonical}

imbedding.

^{The norm in B}

@ 1

^{C is called the}

projective

norm or the trace-norm.

There is an

isometry

^from

(B 01 C)

^{onto LN}

(B’, C),

the Banach _space of linear nuclear

operators

^{from B into C with the trace} norm; this

isometry

is the extension of the

mapping which,

^{to x}

^@

y, associates the

operator

h -~ ~ x, h ~ . Y.

On the other

hand,

^{there is an}

isometry

^{from the dual}

(B 0 ~ 1 C)’

^of

(B 01 C)

^onto

CL(B, C’),

the Banach _space of continuous linear

operators

from B into C’ with the usual _norm; this

isometry

associates to the linear form

f

^{on B}

0~ 1 C,

^the continuous linear

operator f

^defined

by

~ f (x)~ y ~ = f (x~ Y)~

1.4.

QUADRATIC

^INTEGRAL.

For

studying

^second order processes

(see [MaS ], [ChW 2]’

^or

[Wer3])

or

integral

^{w. r. t.}

quadratic

^{Doleans measures}

(see [MeP] ]

^and

[Met 1 D,

it is _necessary to consider

integrals

^{of the}

type XdMY*

^{where M is a}

a-additive function with values in the space of continuous antilinear

operators CL(B’, B")

^{and X and Y are} functions defined on S and with values in B’.

Actually,

^{M can} be considered as a V’ =

(B’ @ 1 B’)’-valued

^additive

function and

f

^{= X}

@

^{Y can be} considered as a .B’

@1 B’ =

^V-valued

function; then,

^the

integral XdMY*

^{= is a}

^special

^{case of}

the

general integral

^{which was discussed in}

1.1; thus,

^{we can use all the}

classical results on such bilinear

integrals.

In the next sections we will

study

^more

precisely

^the

integral

^X

^@ YdM,

when M is a

non-negative

^measure.

"

2. NON-NEGATIVE MEASURE

In this section we shall consider the space V ⁼ B’

0i 1 B’

^{where B is}

a

complex

^Banach space. If X: S -~ B’ and Y: S - B’ ^are measurable

mappings

^then ~ _ _ _ _ _ ~ _ w . __

Annales de l’lnstitut Henri Poincaré - Section B

INTEGRALS RELATED TO STATIONARY PROCESSES 131

is measurable too. We are interested in measures with values in

(B’ 0i B’)’

^{= V’ which are}

non-negative

^{i. e.,} VXeB’ ~0394~03A3

Such measures in natural _way arise in the

theory

^of

stationary

processes

(the

^{measure which}

corresponds

^to the correlation

function)

and in the

theory

^of square

integrable cylindrical martingales (quadratic

^Doleans’

measure),

^see

(§ 8)

^below.

We need the

following definition.

If B ⁼ H is a Hilbert space then ^non-

negative

^measure

E( . )

^with values in the set of all

orthogonal projections

in H is called a

spectral

^{measure. Thus}

^VA, Ai, A~

^{E E.}

2.1. LEMMA.

For each

non-negative

^{measure M} with values in

(B’ Q 1 B’)’

^{there exist}

a Hilbert _space

H,

^an

operator

^{R E}

CL(B’, H)

^{and a}

spectral

^measure

E( . )

such that

Moreover if H is minimal i. _e., H ⁼ then

H,

^{R and}

E( . )

. DEE

are

unique

^{up to}

unitary equivalence.

^{This means that}

ifM(A)

⁼

where

E 1 ( . )

^{is a}

spectral

^{measure in a Hilbert}

Space H and .R1:

^{B’ -}

H 1,

then there exists ^an

unitary operator

^U:

H1

^{H such that R =}

UR1

1

and

UE1 (Ll)

⁼

E(A)U

^{for ‘d0 E E.}

Proof

^{- It} follows from

([W er 2], Prop. 3) by using

^{the fact that}

(B’ 01 1 B’)’

is isometric to CL

(B’, B").

The

triple (H, R, E( . ))

is called dilation of measure

M( . ).

This dilation may be

interpretated

^{in the}

following diagram:

^.

which shows that the measure

M( . )

^is factorized

by

^{the Hilbert}

Space

^H.

2.2. LEMMA.

The Hilbert

Space

^H in the above

diagram

^is

separable

^if

M(S)B’

^is sepa- rable subset of B".

J. PELLAUMAIL AND A. WERON

Proof

^{- Since}

M(S)B’ = R*E(S)RB’

^{c R*M the}

necessity

^{is clear.}

Conversely,

^let

M(S)B’

^be

separable.

^It is sufficient to prove that the

image R(C)

^{c H} of the unit ball C in B’ is

separable.

be a sequence in C such is dense in

M(S)C.

We shall to prove

that ( Rbk}

^{is dense in}

R(C).

^{Let n E}

R(C). 3

^{b E C such}

that Rb ⁼ n.

Choosing

^a

subsequence (

^{such that}

M(S)bkn M(S)b

in B" we have

which

completes

^the

proof.

^{In the last}

equality

^{we use the fact that}

E(S)

⁼

IH

and

consequently M(S)

^{= R*R.}

2.3. REMARK.

Let _vM be the total variation of M. Then

by

^{lemma 1.2 oo iff there}

exists a non

negative

^finite measure p on L such

that ( X I (

for each X E B’.

Measures with such

property

^{was called}

p-bounded

ⁱⁿ

[Mas ].

2.4. LEMMA.

The

following

conditions on measure M ^are

equivalent:

i)

the total variation _vM is infinite :

over all finite collections of vectors from B’

with II

^{1 and all}

parti-

tions of S into a finite number of

disjoint

^sets

in E;

iii)

^{there exists a} B’-valued measurable function X such that

Sup II

¹

and

(X ~ ^X)dM

^{= + 00.}

s

Proof

- From the definition of the total variation of M

immediately

follows that if _vM ⁼ oo then

(ii)

holds. To prove the converse we observe that the norm

of II M(A)!! ( in (B’ 0i 1 B’)’

^is

equal

^to the sup

[M(A) ](X

^@

X),

where the supremum is taken over all vectors X from B’

with II

^1.

Thus

(ii) implies (i).

^Is is easy ^{to see} from the definition of the

integral

f dM

^that

⁽ⁱⁱ⁾

^is

equivalent

^to

(iii).

^Which

completes

^the

proof.

2.5. DEFINITION.

We _say that M is

absolutely

continuous ^{w. r.} t. a

non-negative

^measure /1 on E if for

each x,

y ^E

B’, [M( . ) ](x

@

y)

^« ,u.

Annales de l’lnstitut Henri Poincaré - Section B

INTEGRALS RELATED TO STATIONARY PROCESSES 133

2.6. LEMMA

If for measure M the

image

^{of B’ under}

M(S) : M(S)B’

^{c B" is}

separable,

then there exists a

non-negative

^finite measure p on E such that M is absolu-

tely

continuous w. r. t. J1.

Proof - By

lemma 2 .1

M( . )

^{has a} minimal dilation

M( . )

⁼

R*E( . )R.

Moreover our

assumption

^{on M}

implies by

^{lemma 2.2} that the Hilbert space H in above dilation is

separable.

Observe that we have H ⁼ Let _p be

given by

^{the formula :}

Ael

where

(en)

^{is CONS in H. Let us now use the fact that}

E( . )

^is

spectral

^measure

in

CL(H, H).

^{Thus for}

each n, ~ E(0394)en~2

^is

non-negative

^{finite scalar-}

valued measure on E and

consequently j.1

^is

non-negative

measure on

(S, L)

and ⁼ 1.

Moreover, E( . )en 112

^« j.1. Since

we conclude that

Vh, g

^{E H}

But

for x, y E B’, h = E( . )Rx

^and

g = E( . )Ry belong

^{to H and}

[M(.)](x @ y) = (R *E(. )Rx)(y) = (E2~ . E( )Ry) _ (E( . )h~ g)~

In the last

equalities,

^{we use the fact that for}

spectral

^measure

E2( . )

⁼

E( . ).

Hence

by using

the above facts we obtain that

Vx,

y ^E B’

2.7. EXAMPLE.

Now we

give

^an

example

^of a measure M with values in

(B’ @, B’)’,

where B’ = H is a

separable

^Hilbert space, which is

strongly

^a-additive

but its total variation _v,~ is not a-finite.

Let S =

[0, 1 ]

^x

[0, 1 ],

^{~ =}

a-algebra

of the Borel sets of S, _fi- the

Lebesgue

measure on S and a-the

Lebesgue

measure on

[0, 1 ].

^We

put

where % is the

a-algebra

of the Borel sets on

[0, 1 ].

For each element A ~ 03A3 and for each finite

family ( f , gi)i~l

^{of elements from H x H we define}

Now, m

^{induces an additive}

mapping

^M from E into

(H Q 1 H)’.

^If

A ⁼ B x

C,

^where

B,

^{C c}

[0, 1 ],

^and

then we have the

following inequality

^{for the norm in}

(H Q 1 H)’ :

which shows that the total variation of M is not a-finite.

It remains to prove that M is

strongly

a-additive. Let A be ^an element of ~. It is _easy to see that the. norm of

M(A)

ⁱⁿ

(H Q 1 H)’

^is

equal

^{to the}

supremum of where the _supremum is taken over

all the

pairs ( f, g)

of elements from H

with (

^{1 and}

( ~

^1.

But

by

^the

Cauchy-Schwartz inequality,

^{we have}

Hence

and

consequently

^{M is a} a-additive function on X onto

(H 0 ~ 1 H)’

^with

the

strong topology.

2.8. REMARK.

The above

example

shows that there exists a measure M : ~ --~

(B’ Q 1 B’)’

which is

absolutely

continuous w. r. t. a a-finite _measure, but its total variation _vM is infinite.

3. OPERATOR-VALUED FUNCTIONS AND

QUADRATIC

^INTEGRAL

3.1. NOTATIONS.

In this

section,

^the

assumptions

^{and notations are as in}

^the previous

section 2. More

precisely :

E is ^a

a-algebra

of subsets of

S,

^{B is a} Banach space, M is a

(B’

^(x)

1 B’)’-

valued measure defined on E with total variation _vM, this total variation _vM is finite and a-additive and M’ = ^-2014.

dvM

Moreover we suppose that B’ is

separable

and that G is a Banach _space.

Annales de l’lnstitut Henri Poincaré - Section B

135 INTEGRALS RELATED TO STATIONARY PROCESSES

3.2. ADJOINT OPERATOR

(see [Y os]

p.

196).

Let x be an element of

L(B, G)

^{with domain D dense in}

B; then,

^the

adjoint operator

^{x* is an element}

of L(G’, B’)

^defined

by

for

bED;

3 . 3 . WEAK ^* MEASURABILITY.

Let X be an

L(B, G)-valued

function defined on

S;

^we say that X is

weakly*

measurable

if,

^for every element

(b, g’)

^of

(B

^x

G’), ( X(b), g’ ~

is measurable. ^’

Remark. The situation in this section is more

general

than in the pre- vious ones where we consider the case G ⁼ C and

consequently L(B, G)

^{= B’.}

For

studying

infinite-dimensional stochastic processes in Banach _spaces the

necessity

^{arises of}

using

^such

operator

^functions

(see

^{sections 7 and}

8).

3.4. LEMMA.

Let u be an element of

(B’ 01 1 B’)’, g’

^{be an} element of G’ and v be an

element of

L(B, G)

with domain D dense in B. Let v* be the

adjoint

^{of v.}

Let

(Cn)n>

o be a sequence of elements of

B’,

^dense in B’. Let

Bn

^{the vector}

space

generated by {

^{1, k, n. Let nn a be a linear}

idempotent

contraction from B’ onto

Bn.

^{Then we have :}

Proof: By

^our

assumptions,

~c" ^E

CL(B’, BJ,

^{= C if C E}

Bn

^and

if CeB’. Such

operator

^exists

according

^{to the Hahn-}

Banach theorem.

If we

consider v, g

^{and E >}

0,

^{there exist k > 0 and C E}

B~

^{such that}

~ v*(g) -

^{E ; this}

implies,

^for

every n

^{k :}

Thus {03C0n v*(g)

converges

strongly

^to

v*(g)

^{in B’.}

Now the lemma follows from the

continuity

^{of u and from the}

continuity

of the

mapping (x, y)

^-~

(x

(x)

y)

^{for the « trace norm » on}

(B’ @ 1 B’).

3.5. PRELIMINARY PROPOSITION.

Let X be an

L(B, G)-valued

function defined on S and

weakly*

^measura-

ble. For every element ^s of

S,

^we suppose that the domain of

X(s)

^{is dense}

in B. Let X* be the

adjoint

^{of X.}

Then,

for every

element g

^of

G’,

^{the func-}

tion

M’ { [~*(g) ]°’ ~

^{is a}

(real

^non

negative)

measurable function.

Thus,

the

integral f M’ { [X*( g} ] ° 2 ~ . dvM

is well defined

(finite

^or

infinite).

Proof

^{- Let} o be a sequence of

operators

^as in the above lemma.

This lemma

implies :

But,

^for every

integer

^n,

M’ { [~n ~ X*( g) ] ° 2 ~

^is measurable

according

to the weak*

measurability

^{of X.}

^Then,

^{the same}

^property

^{holds for}

~X*tg)]°2 ~

3.6. DEFINITIONS OF

~g

^AND

^J~.

Let _g be an element of G’.

We say that a

L(B, G)-valued

^{function X defined on S is an element}

of

~~,

^{if :}

i)

^{X is weak*}

measurable,

ii)

^for every element s of

S,

^{the domain}

of X(s)

^{is dense in}

^B, iii) NR(X)

^{+ oo, where}

Moreover,

^{we denote}

by J~

^{the vector} space of all elements X from such that

N~X)

+ oo, where

4. HILBERT SPACE CASE

Here we sketch ^a construction of a

quadratic integral

^{which was}

given

in

[Mas ]. ^H,

^{K are}

separable

^Hilbert space.

Let X be an

L(H, K)

^{valued function on}

S ;

ⁱⁿ

[Mas ] X

^is said measurable if there exists a sequence of

simple

measurable

CL(H, K)

^{fonctions such}

that V E S and for each x E H we have

lim !! X(S)x ))

^{= 0.}

n

Let M be a

non-negative CL(H, H)-valued

^measure on E and M’ be a

weak

Radon-Nikodym

derivative w. r, t. the total variation _vM, which

by

the

assumption

is finite. We note that in

[Mas ] the

author assumed that M is

p-bounded.

^But

by (2 . 3)

^{this is}

equivalent

^{to our}

assumption.

Annales de l’Institut Henri Poincaré - Section B

137

INTEGRALS RELATED TO STATIONARY PROCESSES

4.1. DEFINITION

([Mas]).

The

pair (X, Y)

^{is M}

integrable

^if

i) ^XM,1/2

^{and are}

CL(H, K)

valued and measurable

(see above), ii) the

^function

(XM’1/2).(YM’1/2)*

^is

Bochner vM integrable.

We denote

4. 2. DEFINITION.

~ ~

The functions

X,

^{Y with}

(X - Y)M’ 1 ~2

^{= 0 a. s.}

] are

identified.

From the definition of

L2,M

^{one may see that}

4 . 4 . PROPOSITION

([Mas]).

L2,M

^{is a Hilbert space over the}

ring CL(K, K)

^{with the norm}

and moreover the

simple

^{functions are dense in}

L2,M.

4.5. PROPOSITION.

If H = K,

^{then the measure M has a}

spectral

^{dilation in}

L2,M’

Proof

^{- Let}

E 1( . )

^{be a}

spectral

^{measure in}

L2,M

^{defined as an}

^operator

of

multiplication by

^{the indicator}

1~.

^{Let R} be the inclusion of K into

L2,M.

Then R* is an

orthogonal projection

^{onto K and for each A E L}

4.6. REMARK.

It is

interesting

^to compare the above

integral

^{with one in}

[Met1],

Sect.

IV,

^{which is a}

special

^{case of the}

integral

^{defined in}

(3 . 5) ;

^{we note}

that

by (4 . 3)

the space

L2,M

^and

K) (defined

ⁱⁿ

[Met1],

p.

54)

are very similar.

Vol.

5. STOCHASTIC INTEGRAL 5.1. INTRODUCTION.

In this

section,

^we

give

^a construction of the

integral XdW

^{in a}

^general

context which includes stochastic

integrals

^with

respect

^{to W where W}

can be a

stationary

process

(see ]

^{and or a}

cylindrical martingale (see [Met] ]

^and

(MeP D.

^{A similar}

approach,

^for

stationnary

processes with values in Hilbert _space

only,

^{can be found in}

[Mas ],

^In

our

situation,

^{X is an}

L(B, G)-valued

^function

(or

process

).

The assump- tions on W are as follows :

5.2. NOTATIONS.

.. ’

Let H be a Hilbert _{space and} B and G two Banach spaces where B’ is

separable.

^{Let d be an}

algebra

^of subsets of the set S and E the

6-algebra generated by

d. We denote

by

^{W an} additive function defined on E and with values in

CL(B’, H)

^such

^that,

^for every element

(u, r)

^of

(B’

^x

B’),

and for _every

pair (E, F)

of elements of d such that E n F ⁼

cjJ,

^{we have}

~ W(F)(v) ~H =

^0.

Then,

^for every element

(F,

^u,

v)

^of

(E

^{x B’ x}

B’)

^{we define :}

M is an additive

mapping

^{on d which induces an additive}

mapping

with values in

(B’ (X~ 1 B’)’

^{that we denote also}

by

^M.

We denote

by

vM the total variation of M

(for

^{the norm in}

(B’ Q 1 B’)’).

We suppose that this total variation is finite and a-additive.

In this _case, M can be extended into a

strongly

a-additive

mapping,

defined on L and with values in

(B’ 01 1 B’)’

^{that we} also denote

by

^M.

Then,

^{all the}

assumptions given

^{on M in section 3 are} fulfilled. As

before,

we denote

by

^{M’ the R. N.} derivative of M w. r. t. vM.

5.3. 5i-SIMPLE FUNCTIONS.

We denote

by ~

^{the set of the} functions such that X

= xi .

tel

^lA~~)

^where

(Xi)ieI

^{is a finite}

family

^{of elements of}

L(B, G)

with domain dense in B and

(A(i))iEI

^{is an} associated

family

^of elements of d.

Annales de l’lnstitut Henri Poincaré - Section B

INTEGRALS RELATED TO STATIONARY PROCESSES 139

In this _case, for _every

element g

^of

G’,

^{we define :}

and

XdW is

^an element of

L(G’, H)

^{that we call the}

integral

^{of X}

w. r. t. W.

Of _course,

if,

^for every element i of

I, xt belongs

^to

CL(G’, B’),

^then

( BJ XdW)

^{is an} element of

CL(G’, H) ;

^but

( BJ

^{can be an element}

of

CL(G’, H)

for elements

xt

^{which do not}

belong

^to

CL(G’, B’).

Moreover,

for every element

(E, g)

^{of x}

G’),

^{we define : ,}

Then,

^the

mapping

^E

(EXdW) is

^{an additive}

^mapping

^defined

on A and with values in

L(G’, H).

The

problem

^{is to} extend the

mapping

^X XdW defined on ~ ^to

a

sufficiently large

class of functions.

For this extension the

following elementary

^{lemma is needed :}

S . 4 . LEMMA.

For every element X of ~ and for every element g of

G’,

^{we have :}

Proof.

^{2014 X}

belonging

^to

~,

^{we have X}

= 03A3xi.1A(i).

^{We can suppose}

tel

that the ^{sets are}

pairwise disjoint ;

^{in this case, we have :}

by using

the fact that

W( . )

^is

orthogonal

^on

disjoint

^sets.

Vol. n° 2-1979.

WERON

Then,

^the definition of M

implies

5.5. CONSTRUCTION OF THE INTEGRAL.

The lemma 5-4 above shows

exactly that,

^for every element g of

G’,

the H-valued linear

mapping

^X

H

^XdW

)(g)

^{defined on ~} is continuous for the semi-norm

Ng

^{on 6 c}

~g

^{and the}

^{strong topology}

^{of H.}

Thus,

^this

mapping

^can be extended into an

unique

H-valued linear

mapping

^{defined on} the closure of lff in

~g

^{for the semi-norm}

Ng.

^Conse-

quently

^the

integral XdW

^is

^defined,

^{in a}

^unique

^way, for all the elements of the closure

~203C0

^{of 6 in}

2;,

^{i. e. this}

integral

is defined for all the

L(B, G)-

valued functions X which are

weakly* measurable, X(s)

^{has a domain}

dense in B for _every element s of S and such that

N1[(X)

^{+ 00}

(see

^3-6

above). Moreover,

^{if X}

belongs

^to

6i, XdW

^{is an} element of

CL(G’, H).

6. LOYNES SPACES

Our attention is now directed at spaces ^on which a vector-valued inner

product

^{can be defined.}

Let

Z

be a

complex

^Hausdorff

topological

^vector space

satisfying

^the

following

conditions :

(6 .1 ) ^Z

^{has an} involution : i. ^{e. a}

mapping

^{Z --~}

^{Z +}

^of

Z

to itself with the

properties

(6 . 2)

^{there is a closed} convex cone

P

in

Z

such that

P

n -

P

⁼ 0 ; then

we define a

partial ordering

ⁱⁿ

^Z by writing Z,

^if

Zl - Z,

^E

P ;

Annales de l’lnstitut Henri Poincaré - Section B

INTEGRALS RELATED TO STATIONARY PROCESSES 141

(6 . 3)

^the

topology

ⁱⁿ

^Z

^is

compatible

^{with the}

partial ordenng,

^{in the}

sense that there exists a basic

set { No }

^{of convex}

neighbourhoods

^{of the}

origin

^{such that if x E}

No

^and

x a y a

^{0 then y E}

No ;

(6 . 4)

^ifxe

^P

^{then x =}

x+ ;

(6 . 5) ^Z

^is

complete

^{as a}

locally

^convex space ;

(6 . 6)

^if

... > 0

then the sequence xn is

convergent.

We remark that it is clear that these conditions are satisfied

by

^the

complex

numbers and

by

the space af

all q

x q matrices with the usual

topologies.

Now,

suppose that

H

is a

complex

linear space. A ^vector inner

product

on

H

is a map x, y -

[x, y] ]

^from

^H

^x

H

into an admissible _space

Z (i.

^e.,

satisfying (6.1)-(6.6))

^{with the}

following properties :

for

complex a

^{and b. When a vector inner}

product

is defined on the space

H

there is a natural _{way in} which

H

_{may be} made into a

locally

^convex

^topo- logical

^vector space.

Namely

^{a basic set of}

neighbourhoods

^{of the}

origin { uo }

^{is defined}

^by

(6.11)

DEFINITION.

The space

H

which is

complete

^{in the}

topology

^defined

by (6.10)

^with

the

admissible

_space

Z satisfying

conditions

(6 .1 )-(6 . 6)

will be called a

Loynes

space. This _space was introduced in

[Loy].

A

complex

^Hilbert space and the space of all _p x q matrices with the usual

topologies

^are

simple examples

^of

^Loynes

^spaces.

(6.12)

PROPOSITION

([ChWJ).

Let B be a

complex

^Hilbert space and H ^a

complex

^Hilbert space. Then the space

CL(B, H)

^{with the}

strong operator topology

^{is a}

Loynes

space if we define

A +

as an

operator adjoint

^to

A, P

^{as the set of all}

non-negative operators

^and

[A,C]=C+A

with values in an admissible _space

CL(B, B)

^{with the weak}

operator topo-

logy.

Using

the remark from sect.

1,

^{we have that}

CL(B, B)

is isometric to

(B’ 0i 1 B’)’.

^{We will see below that} the above

Loynes

space

L(B, H)

^is

closely

^{related to}

stationary

processes if H ⁼ and to

2-integrable cylindrical martingales

^{if H = M-the Hilbert} space of real square inte-

grable martingales.

7. STATIONARY PROCESSES

Let H ⁼

L~(Q). By

second order stochastic processes with values in

a Banach space

B, briefly B-process,

^{we mean the}

trajectory Xt

^{in the}

Loynes

spaces

L(B’, H).

^{We refer} the reader to

[Wer1] for

^{the motivation}

and

examples

^{of such} processes. Its correlation function

takes values in the admissible _space

(B’ §) ~ 1 B’)’.

If T is an Abelian group then ^a

B-process

^is

stationary

^{if the function}

of two variables

K(t, s) depends only

^on

(t

^-

s).

^{In the case when T is a}

topological

group, ^we will assume that

K(t)

^is

weakly

continuous i. _e.,

~b~B’,

K(t) (b @ b)

^is continuous. If T is a

locally compact

^Abelian

(LCA) group

^{then we}

have,

^cf.

]

where

E( . )

^{is a}

spectral

^measure in the space

Hx

⁼

Sp { ^{X tb, t}

^E

^T,

^{b E}

^{B’ ~ ,}

defined on the Borel

a-algebra

^{of the} dual group G

and ~ t, g ~

^{-a value}

of the

character g

^{E G at}

point t.

^{In the case T = R we have G = R and}

( t, g ~ - eitg, t, g

^E

R, similary

^{for T =}

Z,

^{G =}

[0, 2n] ] and ~ t, g ~ - eitg,

[0, 2n ].

We note that the

integral (7 .1 )

^{is a}

special

^{case of}

integral

^{w. r. t. an}

orthogonal

^{measure, which was studied in section 5. Indeed}

is an

orthogonal

^measure with values in

CL(B’, H).

^{Here X is}

equal to ~ t,

g

~.

(7.2)

PROPOSITION.

B-process Xt

^is

stationary

^{if its} correlation function has the form

Annales de l’lnstitut Henri Poincaré - Section B

INTEGRALS RELATED TO STATIONARY PROCESSES 143

where M is

non-negative (B’ @ ~ B’)’-valued

^{measure defined on the Borel}

a-algebra BG

^{of the dual group G.}

Proof

^{- If}

Xt

^is

stationary

^{then from the} definition of the correlation function and

(7 .1 )

^{we have}

Thus if we

put

we obtain

(7 . 3). M( . )

^is

non-negative ^because, according

^{to the}

properties

of the

spectral

^measure

E(s) (cf.

^Sect.

2),

^{we have that}

Conversely,

^Let

M( . )

^{be a}

non-negative (B’ @ 1 B’)’-valued

measure on

BG.

Then, by

^{lemme 2.1 there exist a}

Hilbert-space H,

^an

operator

^{R E}

CL(B’, H)

and a

spectral

^measure

E 1 ( . )

^{in H such that}

We may take H ⁼ Let we

put Ut = G

^t,

^{g > E 1 (dg)}

^{for each t E T.}

Then, Ut

^{is a}

unitary operator.

^{Now we define a} process

Xt by

^the

following

formula

Consequently, Xt

^{has the form}

(7.1)

^{and is}

stationary.

(7.4)

^REMARK.

Observe that

W(.)

⁼

E(. )Xe

^{is an}

orthogonal

^{random measure and}

that each

stationary B-process

^{has the}

representation

^as

integral (sto- chastic)

^{w. r. t. an}

orthogonal

^{random measure}

Consequently,

^{the measure M connected with the} correlation function of ^a

stationary B-process (by 7 . 2)

has the form :

Vol. 2-1979.

We note that the above fact is

general

^{and holds for all}

non-negative (B’ @ 1 B’)’

^valued measure on

(S, ~).

^{It follows}

immediately

^{from the}

dilation theorem

(Lemma 2.1).

(7.5)

PROPOSITION.

If

M( . )

^{is a}

non-negative (B’ 01 1 B’)’-valued

measure on

(S, E)

^{then there}

exists a Hilbert _space H and a

orthogonaly CL(B’, H)-valued

^measure

W( . )

^on

(S, L)

^{such that for each A e E}

Moreover,

^{if H} is minimal H ⁼

V : ^W(A)B’}

then H and W are

AeX

determined _up to

unitary equivalence.

(7.6)

^EXAMPLE.

Let B be a Hilbert space. Then stochastic

integral (as

ⁱⁿ

7.4)

is defined

as

mapping

^from

L2,M

^into

H° :

^closed

subspace

^of Hilbert-Schmidt ope- rators

spanned by ^Ai

^E

^{CL(H), ti}

^{E T and N is}

integer

^cf.

^{[MaS ].}

We note

only

that in view of

(2.1)

^and

(4. 5)

this stochastic

integral

^realized

unitary equivalence

^{between two}

spectral

^{dilations of the measure M defined}

by

^the correlation function of the

stationary

process.

8. CYLINDRICAL MARTINGALES

Let

(Q, ff, P,

^{be a}

probabilized

stochastic basis

(see,

^for

example [MeP2 ]).

^Let be the space of all the

complex

square

integrable

^martin-

gales (with respect

^{to this} stochastic

basis) ;

^if

(Ut)teT

^and

(Vt)teT

^{are two such}

martingales,

^{we can define}

For this calar

product ( .,. ) ,

^{~~r is an} Hilbert space.

Let B be ^a Banach _space.

Following Prop. (6.12), CL(B’, ~~~)

^{is a}

Loynes

space. Then, ^an element

W

of

CL(B’, ~~)

^{is called a}

square-integrable cylindrical martingales (see [Met] ]

^or

[MeP1 D.

We define S ⁼ Q x T and we denote

by Z

^the

a-algebra

^{of the}

predictable

Annales de l’lnstitut Henri Poincaré - Section B

145 INTEGRALS RELATED TO STATIONARY PROCESSES

sets

(with respect

^{to the} stochastic basis

given above).

^{We denote}

by d

the

algebra,

^{included in}

E, generated by

^{the sets}

(D

^x

]r, s])

^with

^reT,

s E T and For such an element F ⁼ D ^x

]r, s] ]

^{and for} every element u of

B’,

^{we denote}

by W(F)(u)

^{the real}

martingale

which is defined

by:

Then,

^{W is an} additive function which satisfies all the

properties given

in 5.2 above with H ⁼ -e7.

In this _case, the function

M,

associated to W as in 5-2

above,

^{is the} qua- dratic Doleans measure of the

cylindrical martingale

^{W. If X is a}

weakly*

predictable

process

(with

^{values in}

L(B, G)),

^{X can} be considered as a

weakly*

E-measurable

function;

^{in this} case, XdW is the stochastic

integral

^{of X}

with

respect

^{to W.}

"

For other details and

results,

^see

[Met ] and [MeP 1 ].

REFERENCES

[Bar] ^{R. G.} BARTLE, ^A general ^{bilinear vector} integral. ^Studia Math., ^t. 15, 1956, p. 337-352.

[ChW1] ^S. A. CHOBANYAN and A. WERON, Stationary processes in pseudo-Hilbertian

space (in Russian), ^{Bull. Acad.} Polon., t. 21, 1973, p. 847-854.

[ChW2] ^{S. A. CHOBANYAN and A. WERON, Banach space valued} stationary ^processes

and their linear prediction. Dissertationes Math., ^t. 125, 1975, p. 1-45.

[Dou] ^{R. G. DOUGLAS, On} factoring positive operator functions. J. Math. Mech.,

t. 16, 1966, p. 119-126.

[DuS] ^{N. DUNFORD and J. T. SCHWARTZ, Linear} operators. ^{Part. 1.} Intersciences Publishers, ^1967.

[Loy] ^{R. M. LOYNES, Linear} operator ⁱⁿ Vh-spaces, Trans. Amer. Math. Soc., ^t. 116, 1965, p. 167-180.

[Mas] V. MANDREKAR and H. SALEHI, The square integrability ^of operator-valued

functions with respect ^{to a} non-negative operator ^{valued measure and the} Kolmogorov isomorphism theorem. Indiana University Mathematical Journal.

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[Mas] ^{P. MASANI,} Quasi ^{isometric measures and their} applications. ^{Bull. Amer. Math.}

Soc., ^vol. 76, n° 3, ^1970.

[Met1] ^{M. METIVIER,} Integration ^with respect ^to processes of linear fonctionnals. ^Séminaire

de Probabilité de Rennes, ^1975.

[Met2]

^{M. METIVIER,} Cylindrical martingales and stochastic integrals. ^{Lecture on the} conference. Probability Theory ^{on vector} spaces. Trzebieszowice, ¹⁹⁷⁷ (non published).

[MeP1] M. METIVIER and J. PELLAUMAIL, Cylindrical stochastic integral. ^{Séminaire de} Rennes, 1976.

[MeP2] M. METIVIER and J. PELLAUMAIL, ^{A basic} course on stochastic integration.

Séminaire de Rennes, 1978.

2-1979.

[Pel] ^J. PELLAUMAIL, Sur la dérivation d’une mesure vectorielle. Bull. Soc. Math.

France, ^t. 98, 1970, p. 305-318.

[Tre] ^F. Treves, Topological ^vector spaces, distributions and kernels. Academic Press, 1967.

[Wel] ^{J. N.} WELSH, Constructions of the Hilbert _space L2,M ^for operator ^measures.

Proc. Symp. ^{on vector and} Operator ^{measures. A.} Utah, 1972.

[Wer1] ^{A. WERON,} Prediction theory ^{in Banach} spaces. Lecture Notes in Math., ^vol. 472, 1975, p. 207-227.

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(Manuscrit reçu le 5 ^mars 1979)

Annales de l’Institut Henri Poincaré - Section B